Upper bounds on product and multiplier empirical processes
arXiv:1410.8003
Abstract
We study two empirical process of special structure: firstly, the centred multiplier process indexed by a class , $f \to \left|\sum_{i=1}^N (ξ_i f(X_i) - \E ξf)\right|$, where the i.i.d. multipliers need not be independent of , and secondly, $(f,h) \to \left|\sum_{i=1}^N (f(X_i)h(X_i)-\E f h) \right|$, the centred product process indexed by the classes and . We use chaining methods to obtain high probability upper bounds on the suprema of the two processes using a natural variation of Talagrand's -functionals.